The average of three consecutive even numbers is A. If the next five consecutive even numbers are also included, what will be the average of these eight even numbers in terms of A?

Introduction / Context:
This question tests your understanding of averages and how they change when additional terms are added to a set, especially for sequences with a clear pattern like consecutive even numbers. You are given the average of three consecutive even numbers as A, and then asked to find the new average when the next five consecutive even numbers are included. This combines knowledge of arithmetic progressions and the concept of mean.

Given Data / Assumptions:

- The first three numbers are consecutive even integers. - Their average is A. - The next five consecutive even numbers following these three are added. - We now have a total of eight consecutive even numbers. - We must express the new average in terms of A.

Concept / Approach:
Consecutive even numbers form an arithmetic progression with common difference 2. The average of equally spaced numbers is the middle term (if the count is odd) or the average of the two middle terms (if the count is even). We use the fact that for three consecutive even numbers, the middle number is the average, and then describe all eight numbers relative to that middle number. Finally, we compute the average of the new set using the first and last terms or by summing relative offsets.

Step-by-Step Solution:
Step 1: Let the three consecutive even numbers be n - 2, n and n + 2. Step 2: The average of these three numbers is (n - 2 + n + n + 2) / 3 = 3n / 3 = n. Step 3: We are told that this average is A, so n = A. Step 4: The next five consecutive even numbers are obtained by adding 2 successively: A + 4, A + 6, A + 8, A + 10 and A + 12. Step 5: Altogether, the eight numbers are: A - 2, A, A + 2, A + 4, A + 6, A + 8, A + 10 and A + 12. Step 6: These eight numbers form an arithmetic progression with first term A - 2 and last term A + 12, and common difference 2. Step 7: The average of an arithmetic progression is equal to the average of the first and last terms. Step 8: Compute the new average: ((A - 2) + (A + 12)) / 2 = (2A + 10) / 2 = A + 5.

Verification / Alternative check:
You can verify the formula with an example. Suppose the original three numbers are 4, 6 and 8. Their average is (4 + 6 + 8) / 3 = 6, so A = 6. The next five even numbers are 10, 12, 14, 16 and 18. The full list of eight numbers is 4, 6, 8, 10, 12, 14, 16, 18. Their sum is 88, and the average is 88 / 8 = 11, which equals A + 5 = 6 + 5. This confirms the derived expression.

Why Other Options Are Wrong:
The options A + 3, A + 4 and A + 7 come from incorrect assumptions about how far the overall center shifts when more numbers are added. For instance, simply adding a fixed constant or only looking at the next number ignores the influence of the entire group of eight. Only A + 5 matches the exact calculation of the new mean using first and last terms.

Common Pitfalls:
A common mistake is to assume that the new average is just the average of A and the last new number, which is not correct for combined sets. Another pitfall is to miscount the number of terms or forget that the original average A equals the middle number of the first three. Carefully listing the actual numbers in terms of A and using the arithmetic progression average formula avoids these errors.

Final Answer:
The average of all eight consecutive even numbers is A + 5, which corresponds to option C.